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Heisenberg-Euler and the quantum dilogarithm

Gerald V. Dunne

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Phys. Rev. D 113, 085018 – Published 22 April, 2026

DOI: https://doi.org/10.1103/ly3t-zp5l

Abstract

A dispersion integral representation of the Heisenberg-Euler QED effective Lagrangian is derived, with Faddeev’s quantum dilogarithm as a generalized Borel kernel. The nonperturbative imaginary part of the effective Lagrangian is expressed as the quantum dilogarithm, while the real part has the form of a dispersion integral involving both the quantum dilogarithm and its modular dual, a manifestation of electromagnetic duality. The Heisenberg-Euler effective Lagrangian generates all one-loop QED scattering amplitudes in a constant external field, with the Lorentz invariants of the constant background electromagnetic field playing the role of the Mandelstam variables in conventional QED dispersion theory.

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References (46)

  1. W. Heisenberg and H. Euler, Folgerungen aus der diracschen theorie des positrons, Z. Phys. 98, 714 (1936).
  2. V. Weisskopf, Über die Elektrodynamik des Vakuums auf Grund der Quantentheorie des Elektrons, Kong. Dans. Vid. Selsk. Math-fys. Medd. XIV No. 6 (1936); reprinted in V. WeisskopfQuantum Electrodynamics, edited by J. Schwinger (Dover, New York, 1958).
  3. A. I. Nikishov, Pair production by a constant external field, Zh. Eksp. Teor. Fiz. 57, 1210 (1969) [Sov. Phys. JETP 30, 660, 1970].
  4. W. Dittrich and M. Reuter, Effective Lagrangians in Quantum Electrodynamics (Springer, New York, 1985).
  5. W. Dittrich and H. Gies, Probing the quantum Vacuum. Perturbative Effective Action Approach in Quantum ElectrodyNamics and Its Application, Springer Tracts Mod. Phys. Vol. 166 (Springer, New York, 2000), pp. 1–241.
  6. G. V. Dunne, Euler-Heisenberg effective Lagrangians: Basics and extensions, in Ian Kogan memorial collection, From Fields to Strings: Circumnavigating Theoretical Physics, Vol. I, edited by M. A. Shifman et al. (World Scientific, Singapore, 2004).
  7. M. D. Schwartz, Quantum Field Theory and the Standard Model (Cambridge University Press, Cambridge, England, 2014).
  8. J. S. Schwinger, Quantum electrodynamics. I. A covariant formulation, Phys. Rev. 74, 1439 (1948); Quantum electrodynamics. 2. Vacuum polarization and self-energy, 75, 651 (1948).
  9. J. S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82, 664 (1951).
  10. R. P. Feynman, Space—time approach to quantum electrodynamics, Phys. Rev. 76, 769 (1949); The theory of positrons, 76, 749 (1949).
  11. R. P. Feynman, Mathematical formulation of the quantum theory of electromagnetic interaction, Phys. Rev. 80, 440 (1950).
  12. Y. Nambu, The use of the proper time in quantum electrodynamics I, Prog. Theor. Phys. 5, 82 (1950).
  13. C. Morette, On the definition and approximation of Feynman’s path integrals, Phys. Rev. 81, 848 (1951).
  14. H. J. Silverstone, High-order perturbation theory and its application to atoms in strong fields, in Atoms in Strong Fields (NATO Science Series B: 212), edited by C. A. Nicolaides, C. W. Clark, and M. H. Nayfeh (Springer, New York, 1990).
  15. V. Yakimenko, S. Meuren, F. Del Gaudio, C. Baumann, A. Fedotov, F. Fiuza, T. Grismayer, M. J. Hogan, A. Pukhov and L. O. Silva et al., Prospect of studying nonperturbative QED with beam-beam collisions, Phys. Rev. Lett. 122, 190404 (2019).
  16. P. H. Bucksbaum, G. V. Dunne, F. Fiuza, S. Meuren, M. E. Peskin, D. A. Reis, G. Torgrimsson, G. White, and V. Yakimenko, Understanding the fully non-perturbative strong-field regime of QED, in Letter of Intent to Snowmass 2021 Theory Frontier (2021), https://dunne.physics.uconn.edu/non-perturbative-qed/.
  17. H. Abramowicz, U. Acosta, M. Altarelli, R. Aßmann, Z. Bai, T. Behnke, Y. Benhammou, T. Blackburn, S. Boogert and O. Borysov et al., Conceptual design report for the LUXE experiment, Eur. Phys. J. Special Topics 230, 2445 (2021).
  18. H. G. Rinderknecht, E. Dill, A. J. MacLeod, B. King, K. Sow, S.-W. Bahk, I. A. Begishev, F. Karbstein, Jörg Schreiber, M. Zepf, and A. Di Piazza, On measuring stimulated photon–photon scattering using multiple ultraintense lasers, Phys. Plasmas 32, 083301 (2025).
  19. G. Sarri, B. King, T. Blackburn, A. Ilderton, S. Boogert, S. S. Bulanov, S. V. Bulanov, A. Di Piazza, L. Ji, F. Karbstein et al., Input to the European strategy for particle physics: strong-field quantum electrodynamics, Eur. Phys. J. Plus 140, 1151 (2025).
  20. A. Di Piazza, C. Muller, K. Z. Hatsagortsyan, and C. H. Keitel, Extremely high-intensity laser interactions with fundamental quantum systems, Rev. Mod. Phys. 84, 1177 (2012).
  21. A. Gonoskov, T. G. Blackburn, M. Marklund, and S. S. Bulanov, Charged particle motion and radiation in strong electromagnetic fields, Rev. Mod. Phys. 94, 045001 (2022).
  22. A. Fedotov, A. Ilderton, F. Karbstein, B. King, D. Seipt, H. Taya, and G. Torgrimsson, Advances in QED with intense background fields, Phys. Rep. 1010, 1 (2023).
  23. S. L. Lebedev and V. I. Ritus, Dispersion representation for the Lagrange function of an intense field, Pis’ma Zh. Eksp. Teor. Fiz. 28, 298 (1978) [JETP Lett. 28, 274 (1978)].
  24. L. Faddeev and R. Kashaev, Quantum dilogarithm, Mod. Phys. Lett. A 09, 427 (1994).
  25. A. N. Kirillov, Dilogarithm identities, Prog. Theor. Phys. Suppl. 118, 61 (1995).
  26. D. Zagier, The dilogarithm function, in Frontiers in Number Theory, Physics, and Geometry II, edited by P. Cartier et al. (Springer, Heidelberg, 2007).
  27. V. V. Fock and A. B. Goncharov, The quantum dilogarithm and representations of quantum cluster varieties, Inventiones Mathematicae 175, 223 (2009).
  28. R. M. Kashaev and T. Nakanishi, Classical and quantum dilogarithm identities, SIGMA 7, 102 (2011).
  29. S. Garoufalidis and R. Kashaev, Resurgence of Faddeev’s quantum dilogarithm, in Topology and Geometry, edited by A. Papadopoulos (European Mathematical Society, 2021), pp. 257–272.
  30. E. Witten, On S-duality in Abelian gauge theory, Sel. Math. Sov. 1, 383 (1995).
  31. J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields (McGraw Hill, New York, 1965).
  32. R. J. Eden, P. V. Landshoff, D. I. Olive, and J. C. Polkinghorne, The Analytic S-Matrix (Cambridge University Press, Cambridge, England, 1966).
  33. S. Caron-Huot, M. Giroux, H. S. Hannesdottir, and S. Mizera, Crossing beyond scattering amplitudes, J. High Energy Phys. 04 (2024) 060.
  34. Y. Hatsuda and K. Okuyama, Resummations and non-perturbative corrections, J. High Energy Phys. 09 (2015) 051.
  35. S. R. Valluri, D. R. Lamm, and W. J. Mielniczuk, Applications of the representation of the Heisenberg-Euler Lagrangian by means of special functions, Can. J. Phys. 71, 389 (1993); U. D. Jentschura, H. Gies, S. R. Valluri, D. R. Lamm, and E. J. Weniger, QED effective action revisited, 80, 267 (2002).
  36. R. B. Dingle, The fermi-dirac integrals, Applied Scientific Research, Section B 6, 225 (1957); The Bose-Einstein integrals, 6, 240 (1957).
  37. See for example, Appendices D and E in: R. K. Pathria and P. D. Beale, Statistical Mechanics (Academic Press, New York, 2022).
  38. See Entry 19 in Chapter 14 of B. C. Berndt, Ramanujan’s Notebooks. Part II (Springer, Berlin, 1989).
  39. D. Gang and Y. Hatsuda, S-duality resurgence in SL(2) Chern-Simons theory, J. High Energy Phys. 07 (2018) 053.
  40. q-Pochhammer function: dlmf.nist.gov/17.2 https://dlmf.nist.gov/17.2.
  41. G. V. Dunne and C. Schubert, Worldline instantons and pair production in inhomogeneous fields, Phys. Rev. D 72, 105004 (2005).
  42. G. V. Dunne, Q. h. Wang, H. Gies, and C. Schubert, Worldline instantons. II. The fluctuation prefactor, Phys. Rev. D 73, 065028 (2006).
  43. G. V. Dunne and T. Hall, On the QED effective action in time dependent electric backgrounds, Phys. Rev. D 58, 105022 (1998).
  44. G. V. Dunne and Z. Harris, Resurgence of the effective action in inhomogeneous fields, Phys. Rev. D 107, 065003 (2023).
  45. V. I. Ritus, Lagrangian of an intense electromagnetic field and quantum electrodynamics at short distances, Zh. Eksp. Teor. Fiz. 69, 1517 (1975) [Sov. Phys. JETP 42, 774 (1976)].
  46. I. Huet, M. Rausch De Traubenberg, and C. Schubert, Three-loop Euler-Heisenberg Lagrangian in 1+1 QED, part 1: Single fermion-loop part, J. High Energy Phys. 03 (2019) 167.

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Phys. Rev. D 113, 085018– Published 22 April, 2026
  • Received 18 December 2025
  • Accepted 31 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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